Entropy stable discontinuous Galerkin method for Euler equations using non-conservative variables

A conservative version of the entropy stable discontinuous Galerkin method for Euler equations is proposed in variables: density, momentum density, and pressure. A special difference approximation in time, conservative in total energy is constructed for the equation describing the dynamics of the average pressure in a finite element. The entropic inequality and the requirements for the monotonicity of the numerical solution are ensured by a special slope limiter. The method developed has been successfully tested on a number of model gasdynamic problems. In particular, the quality of numerical calculation the specific internal energy has been significantly improved for the Einfeldt problem.